In sub-parts (i) and (ii) choose the correct option and in sub-parts (iii) to (v), answer the questions as instructed.
For the data (1, 2), (2, 4), (3, 5) find the slope of regression equation x on y.

Answer

To find the slope of the regression equation of \( x \) on \( y \) for the given data points \((1, 2), (2, 4), (3, 5)\), we will follow these steps:

### Step 1: Organize the Data
First, we will organize the data into a table format to calculate the necessary values.

| \( x \) | \( y \) | \( xy \) | \( y^2 \) |
|---------|---------|----------|-----------|
| 1 | 2 | 2 | 4 |
| 2 | 4 | 8 | 16 |
| 3 | 5 | 15 | 25 |

### Step 2: Calculate Summations
Now, we will calculate the necessary summations:

- \( \sum x = 1 + 2 + 3 = 6 \)
- \( \sum y = 2 + 4 + 5 = 11 \)
- \( \sum xy = 2 + 8 + 15 = 25 \)
- \( \sum y^2 = 4 + 16 + 25 = 45 \)

### Step 3: Use the Formula for Slope
The formula for the slope \( b_{xy} \) of the regression line of \( x \) on \( y \) is given by:

\[ b_{xy} = \frac{n \sum xy - \sum x \sum y}{n \sum y^2 - (\sum y)^2} \]

Where:
- \( n \) is the number of data points.

In this case, \( n = 3 \).

### Step 4: Substitute the Values
Now we will substitute the calculated values into the formula:

\[ b_{xy} = \frac{3 \cdot 25 - 6 \cdot 11}{3 \cdot 45 - 11^2} \]

Calculating the numerator:

\[ 3 \cdot 25 = 75 \]
\[ 6 \cdot 11 = 66 \]
So, the numerator becomes:

\[ 75 - 66 = 9 \]

Now calculating the denominator:

\[ 3 \cdot 45 = 135 \]
\[ 11^2 = 121 \]
So, the denominator becomes:

\[ 135 - 121 = 14 \]

### Step 5: Calculate the Slope
Now we can calculate the slope:

\[ b_{xy} = \frac{9}{14} \approx 0.642857 \]

### Final Answer
Thus, the slope of the regression equation of \( x \) on \( y \) is:

\[ b_{xy} \approx 0.6429 \]

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